Saeid Alikhani

Dominating Sets and Domination Polynomials of Paths

Let be a simple graph. A set is a dominating set of , if every vertex in is adjacent to at least one vertex in . Let be the family of all dominating sets of a path with cardinality , and let . In this paper, we construct , and obtain a recursive formula for . Using this recursive formula, we consider the polynomial , which we call domination polynomial of paths and obtain some properties of this polynomial.

On the Domination Polynomial of Some Graph Operations

Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . Every root of is called the domination root of . In this paper, we study the domination polynomial of some graph operations.

On the graphs with four distinct domination roots

The domination polynomial of a graph G of order n is the polynomial , where d(G, i) is the number of dominating vertex sets of G with cardinality i. A root of D(G, x) is called a domination root of G. In this paper, we characterize graphs with exactly four distinct domination roots .

Domination Polynomials of k-Tree Related Graphs

Let G be a simple graph of order n. The domination polynomial of G is the polynomial , where d(G, i) is the number of dominating sets of G of size i and γ(G) is the domination number of G. In this paper, we study the domination polynomials of several classes of k-tree related graphs. Also, we present families of these kinds of graphs, whose domination polynomials have no nonzero real roots.

Randić energy of specific graphs

Let G be a simple graph with vertex set V ( G ) = { v 1 , v 2 , ? , v n } . The Randic matrix of G, denoted by R(G), is defined as the n × n matrix whose (i, j)-entry is ( d i d j ) - 1 2 if vi and vj are adjacent and 0 for another cases. Let the eigenvalues of the Randic matrix R(G) be ?1 ? ?2 ? ??? ? ?n which are the roots of the Randic characteristic polynomial ? i = 1 n ( ? - ? i ) . The Randic energy RE of G is the sum of absolute values of the eigenvalues of R(G). In this paper, we compute the Randic characteristic polynomial and the Randic energy for specific graphs.

Graphs Whose Certain Polynomials Have Few Distinct Roots

Let be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of . Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded.

Algebraic Integers as Chromatic and Domination Roots

Let be a simple graph of order and . A mapping is called a -colouring of if whenever the vertices and are adjacent in . The number of distinct -colourings of , denoted by , is called the chromatic polynomial of . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . Every root of and is called the chromatic root and the domination root of , respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.

Distinguishing number and distinguishing index of some operations on graphs

Abstract The distinguishing number (index) D(G) (Dʹ(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. We examine the effects on D(G) and Dʹ(G) when G is modified by operations on vertex and edge of G. Let G be a connected graph of order n ≥ 3. We show that –1 ≤ D(G – v) – D(G) ≤ D(G), where G – v denotes the graph obtained from G by removal of a vertex v and all edges incident to v and these inequalities are true for the distinguishing index. Also we prove that |D(G – e) – D(G)| ≤ 2 and –1 ≤ Dʹ (G – e) – Dʹ(G) ≤ 2, where G – e denotes the graph obtained from G by simply removing the edge e. Finally we consider the vertex contraction and the edge contraction of G and prove that the edge contraction decrease the distinguishing number (index) of G by at most one and increase by at most 3D(G) (3Dʹ(G)).

THE k-INDEPENDENT GRAPH OF A GRAPH

Let

On the Structure of Dominating Graphs

G=(V,E)